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Scalar Science
What is it?
Part III
Presented by Doug Bundy
January 17, 2007
Larsonian Science
• New science
– Based on unification of all things
– New ideas call for new approach
• Includes some of Newtonian science
– Changes some foundational mathematics and
physical concepts, but not all
“Whole New Science”
• Smolin
– “Great unifications become the founding ideas
on which whole new sciences are erected.”
• What are the RST’s “founding ideas?”
– Redefinition of space, time, energy and matter
• What is conserved?
– Motion is conserved
Three Dimensions of Magnitude
Chart of Motion
Change
of Units
Change of
Positions
Change of
Intervals
Change of
Scales
10
20
30
40
Point unit
1
11
1
1
21
1
31
41
4
Line unit
1
12
22
4
3
2
32
42
4
1
9
4
16
8
Area unit
12
13
23
33
43
16
Volume unit
16
1
8
16
64
27
16
16
Conservation of Motion
• Four bases of motion
–
–
–
–
M1 = abstract magnitudes
M2 = change of position magnitudes
M3 = change of interval magnitudes
M4 = change of scale magnitudes
• Three dimensions of motion
–
–
–
–
Mn0 = 0D magnitudes
Mn1 = 1D magnitudes
Mn2 = 2D magnitudes
Mn3 = 3D magnitudes
Calculus
• Defines velocity in context of infinitely
divisible continuum.
• Velocity can vary in arbitrary distances
– Thus, must take delta to the limit
• Applies to M2, change of position motion
• Does not apply directly to M3 and M4 motion
The Newtonian Concept of Force
•
•
•
•
Ultimately a quantity of change of motion
A quantity of acceleration
F = ma, or a scalar times a component vector
Has evolved to fundamental, autonomous, status
–
–
–
–
Electromagnetic “charge”
Weak nuclear “charge”
Strong nuclear “charge”
Gravitational “charge”
Dimensions of Force and
Acceleration
• Force dimensions are energy per unit space
– t/s * 1/s = t/s2
• Acceleration dimensions are velocity per unit time
– s/t * 1/t = s/t2
• Energy is scalar & velocity is vectorial
– As “quantity of acceleration,” force should have
dimensions of acceleration
– Both should be scalar, not vectorial
– (t/s * 1/s) = (t/s)3 + (s/t * 1/t) = t/s2,
• Force = energy per scalar unit of space
• Acceleration = velocity per scalar unit of time.
The Newtonian Concept of Work
• Work divides energy into two concepts
– Potential energy (no direction)
– Kinetic energy (directed energy, or energy of motion (vector).
• Thus, force dimensions become energy dimensions
through displacement (change of position motion)
– Kinetic energy is force by distance
– t/s2 * s = t/s
– If mass is displaced (moves), work is performed and the
potential energy of force (energy per unit space) is
transformed into the kinetic energy of force (energy per unit
space times displacement, or length) in a given direction.
• Forms basis of analysis in Newtonian science
Newtonian Principles of Analysis
• Based on concepts of M2 motion
– Vectors
– Vector spaces
– Functions in vector spaces
• Focuses on Analysis of vectors in vector spaces
– Spaces of vectors are linear spaces
– Thus, in a given vector space
• Vectors can be added together
• Vectors can be multiplied by scalars
– Need not be limited to geometric spaces (visualizable in
three dimensions), but may also be abstract spaces
• Represents vectors with complex numbers
– Opens whole new world of possibilities
– Transforms vectors into scalars!
The Vector Space of Transformations
• Linear operators
– Operator = transform of functions
• Example: differential operator (f(x) -> f’(x))
• An operator is a symbol that tells you to do something
with whatever follows the symbol
– Linear operator
• Satisfies two conditions: An operator O is said to be
linear if, for every pair of functions f and g and scalar s ,
– O(f+g) = Of +Og and
– O(sf) = sOf
• In other words, distributive (ordered) functions (functions
compatible with the addition and scalar multiplication)
Definitions and Dimensions
• Vis morte, or dead force of mass (inertia)
• Vis viva, or live force of motion
– Became energy per Leibniz’s idea (E = mv2) in conservation of energy
– Initially was momentum per Newton’s idea (p = mv) in conservation of
momentum
• Dimensions of momentum are energy squared
– p = mv = (t/s)3 * s/t = (t/s)2
• Dimensions of energy are mass times velocity squared
– E = mv2 = (t/s)3 * (s/t)2 = t/s
• Dimensions of mass are momentum times energy
– m = p * E = (t/s)2 * t/s = (t/s)3
• Thus, energy, momentum and mass are 1, 2, and 3
dimensional magnitudes of inverse velocity
– Energy (t/s); Momentum (t/s)2, and Mass (t/s)3
Conservation Law Sans Force
Concept
• Conservation of motion
– Two forms of motion
• Velocity (s/t)
• Inverse velocity (t/s)
– Two modes of motion
• Translational (unbounded)
• Vibrational (bounded)
• Conservation of direction
– M2 is motion in one direction (line)
– M3 is motion in two simultaneous directions (area)
– M4 is motion in three simultaneous directions (volume)
• Conservation of dimension
Draft Plan for Erecting New Science
1. Compare and contrast with previous science
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•
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View in context of chart of motions
Identify where new unification simplifies
Document findings
2. Identify systematic tools to use
•
•
•
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Reciprocal System of Mathematics
Chart of Motion
World line charts
Progression Algorithms (PAs)
3. Synthesize effective analysis procedure
•
Conservation of motion analyzed in terms of form, mode,
and dimension
Learning from Newtonian Science
• Examine history of mechanical analysis
– Vector analysis
• Examine history of quantum mechanical analysis
– Functional analysis
• Examine history of mathematical development
– Differential calculus
– Linear analysis
– Operator and group theory
• Translate into lessons learned
Mechanical Analysis
• Revisit physical concepts
– Energy, momentum, mass
– Force and acceleration
– Conservation and invariance (symmetry)
• Look for clarification of mathematics
– Number systems
– Discrete principles vs. continuum principles
– Meaning and use of complex numbers
Quantum Mechanical Analysis
• Examine changes in mechanical concepts
– Rotation and angular momentum
– Discrete energy viz-a-viz potential/kinetic concept
– Role of potential energy in wave equation
• Look for mathematical meaning of rotation
– Complex numbers and vector spaces
– Quantum phase and renormalization
– Meaning of non-commutative mathematics
• Master key concepts of standard model
– Gauge principle
– Lie groups and Lie algebra
– Higgs potential and Higgs Boson
What We’ve Learned So Far
• Potential, kinetic, & total energy of pendulum
– Total energy conserved as potential, kinetic energy transform in SHM
• Rotation is special case of pendulum SHM
– With no gravitational field, rotation is analog of pendulum SHM
– Wave equation maps rotation to pendulum dynamics
– Same dynamics without reversals of direction
• Key is complex number solution to wave equation (WE)
–
–
–
–
WE reduces to d2ψ(x)/dx2 +/- k2ψ(x) = 0, with + and - solutions
Positive solution has form of SHM, but
Negative solution has form of rotation expressed as complex number
Only discrete solutions are allowed
• Schrödinger equation is QM version of conservation of energy
Lessons Learned (cont)
• Gauge principle (Yang-Mills) is key to making principle of
energy conservation work using wave equation.
– Provides a way to change phase (rotate system) without rotating
every point in the universe; that is, it allows replacing global
invariance (symmetry) with local invariance (symmetry).
There is an infinite number
of points on the unit circle.
z
bi
z = a + bi
There is a complex number,
z = a + bi, for every point
between 1 and –1 on unit
circle.
a
These rotations form a vector space, using one,
complex, dimension, forming the basis of the U(1) Lie
group, corresponding to 2D geometric rotations.
There is an infinite number of
unit lines, (a2 + b2)1/2 = 1,
corresponding to the z points.
Lessons Learned (cont)
• In two, complex, dimensions, the vector space
forms the basis for the SU(2) Lie group, which
corresponds to 3D geometric rotations.
There is an infinite number
of points on the unit circle.
There is a complex number,
z = a + bi, for every point
between 1 and –1 on unit
circle.
z
bi
z = a + bi
z’ z’ = a’ + b’i
a
These rotations form a vector space, using two,
complex, dimensions, forming the basis of the SU(2)
Lie group, corresponding to 3D geometric rotations.
There is an infinite number of
unit lines, (a2 + b2)1/2 = 1,
corresponding to the z points.
Lessons Learned (cont)
• SU(2) vis-a-vis R(3)
– The SU(2) Lie group and Lie algebra correspond to the 3D
geometric rotation group
– Except it takes 720 degree rotation (4pi) to return to starting point,
not 360 degrees (2pi)! (The story of spin)
• Nobody knows why (do we?)
– Hint: M3 cycle is a 720 degree cycle!
Bruce Schumm writes (Deep Down Things, 2004):
“What is spin and this oddly construed spin-space in which it
lives? On the one hand it is quite real [corresponds to angular
momentum]. On the other hand, a particle with no spatial extent
[electron is point particle] shouldn’t possess angular momentum
[or] have to be rotated through 720 degrees to return the particle to
its original state. We don’t really have a clue about the physical
origin of spin...”
Lessons Learned (cont)
• The use of phase change in one, complex,
dimension, enables application of conservation of
energy in terms of phase changes (rotations) in
U(1) group with reference to the electromagnetic
interaction.
• The isospin concept extends the idea of phase
changes and conservation in SU(2) to the weak
nuclear force, where phase changes in two,
complex, dimensions leads to the weak
interaction, which is a short-range force permitting
the prediction of radioactive decay events.
Conclusions from Lessons Learned
• The concept of M2, (change of position) motion, has
been utilized to
– attempt the description of magnitudes of M3 (change of
interval) and M4 (change of scale) motion
• The attempt has been only partially successful
– As far as it goes, it’s extremely accurate, but it’s incomplete
without the Higgs potential and Higgs boson
• There are many flags to alert us that chart of motion
will shed light on these problems
– Natural explanation of spin
– Insight into rotation – change of interval correspondence
– Clarification of point particles and charges (distribution of
charge on electron)
Predictions
• We will be able to bridge to Newtonian
science, from Larsonian science. It’s only a
matter of time and resources
• Once there, LST physicists will take serious
look at our new science
• Those who jump on board now will be glad
they did later.